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Properties

Label 1125.4.a.q
Level $1125$
Weight $4$
Character orbit 1125.a
Self dual yes
Analytic conductor $66.377$
Analytic rank $1$
Dimension $12$
Inner twists $2$

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Show commands: Magma / Pari/GP / SageMath

Newspace parameters

Copy content comment:Compute space of new eigenforms
 
Copy content gp:[N,k,chi] = [1125,4,Mod(1,1125)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)
 
Copy content magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1125.1"); S:= CuspForms(chi, 4); N := Newforms(S);
 
Copy content sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1125, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 4, names="a")
 
Level: \( N \) \(=\) \( 1125 = 3^{2} \cdot 5^{3} \)
Weight: \( k \) \(=\) \( 4 \)
Character orbit: \([\chi]\) \(=\) 1125.a (trivial)

Newform invariants

Copy content comment:select newform
 
Copy content sage:traces = [12,0,0,54,0,0,0,0,0,0,-170] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)
 
Copy content gp:f = lf[1] \\ Warning: the index may be different
 
Self dual: yes
Analytic conductor: \(66.3771487565\)
Analytic rank: \(1\)
Dimension: \(12\)
Coefficient field: \(\mathbb{Q}[x]/(x^{12} - \cdots)\)
Copy content comment:defining polynomial
 
Copy content gp:f.mod \\ as an extension of the character field
 
Defining polynomial: \( x^{12} - 75x^{10} + 2040x^{8} - 24475x^{6} + 126325x^{4} - 223000x^{2} + 122000 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2^{2}\cdot 5^{8} \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

$q$-expansion

Copy content comment:q-expansion
 
Copy content sage:f.q_expansion() # note that sage often uses an isomorphic number field
 
Copy content gp:mfcoefs(f, 20)
 

Coefficients of the \(q\)-expansion are expressed in terms of a basis \(1,\beta_1,\ldots,\beta_{11}\) for the coefficient ring described below. We also show the integral \(q\)-expansion of the trace form.

\(f(q)\) \(=\) \( q + \beta_1 q^{2} + (\beta_{2} + 5) q^{4} + ( - \beta_{10} - 2 \beta_1) q^{7} + (\beta_{10} + \beta_{9} + 5 \beta_1) q^{8} + ( - \beta_{11} + \beta_{3} - 2 \beta_{2} - 15) q^{11} + ( - \beta_{10} - \beta_{9} + \cdots - 3 \beta_1) q^{13}+ \cdots + (5 \beta_{10} + 10 \beta_{9} + \cdots - 14 \beta_1) q^{98}+O(q^{100}) \) Copy content Toggle raw display
\(\operatorname{Tr}(f)(q)\) \(=\) \( 12 q + 54 q^{4} - 170 q^{11} - 290 q^{14} + 258 q^{16} + 152 q^{19} - 360 q^{26} - 630 q^{29} - 454 q^{31} - 780 q^{34} - 1260 q^{41} - 3340 q^{44} - 640 q^{46} - 16 q^{49} - 4410 q^{56} - 2970 q^{59} - 256 q^{61}+ \cdots + 2200 q^{94}+O(q^{100}) \) Copy content Toggle raw display

Basis of coefficient ring in terms of a root \(\nu\) of \( x^{12} - 75x^{10} + 2040x^{8} - 24475x^{6} + 126325x^{4} - 223000x^{2} + 122000 \) : Copy content Toggle raw display

\(\beta_{1}\)\(=\) \( \nu \) Copy content Toggle raw display
\(\beta_{2}\)\(=\) \( \nu^{2} - 13 \) Copy content Toggle raw display
\(\beta_{3}\)\(=\) \( ( 4\nu^{10} - 325\nu^{8} + 9550\nu^{6} - 121250\nu^{4} + 602625\nu^{2} - 609340 ) / 13680 \) Copy content Toggle raw display
\(\beta_{4}\)\(=\) \( ( -4\nu^{11} + 325\nu^{9} - 9550\nu^{7} + 121250\nu^{5} - 602625\nu^{3} + 595660\nu ) / 13680 \) Copy content Toggle raw display
\(\beta_{5}\)\(=\) \( ( -83\nu^{11} + 6359\nu^{9} - 176360\nu^{7} + 2105965\nu^{5} - 9626325\nu^{3} + 7453100\nu ) / 136800 \) Copy content Toggle raw display
\(\beta_{6}\)\(=\) \( ( -7\nu^{11} + 507\nu^{9} - 12960\nu^{7} + 137945\nu^{5} - 546225\nu^{3} + 343300\nu ) / 7600 \) Copy content Toggle raw display
\(\beta_{7}\)\(=\) \( ( 13\nu^{10} - 928\nu^{8} + 23770\nu^{6} - 261965\nu^{4} + 1140510\nu^{2} - 1039000 ) / 13680 \) Copy content Toggle raw display
\(\beta_{8}\)\(=\) \( ( -104\nu^{10} + 7595\nu^{8} - 195290\nu^{6} + 2066650\nu^{4} - 7866375\nu^{2} + 5422100 ) / 68400 \) Copy content Toggle raw display
\(\beta_{9}\)\(=\) \( ( -253\nu^{11} + 18547\nu^{9} - 485620\nu^{7} + 5421695\nu^{5} - 23663325\nu^{3} + 20619100\nu ) / 136800 \) Copy content Toggle raw display
\(\beta_{10}\)\(=\) \( ( 253\nu^{11} - 18547\nu^{9} + 485620\nu^{7} - 5421695\nu^{5} + 23800125\nu^{3} - 23491900\nu ) / 136800 \) Copy content Toggle raw display
\(\beta_{11}\)\(=\) \( ( -71\nu^{10} + 4985\nu^{8} - 122060\nu^{6} + 1230925\nu^{4} - 4770375\nu^{2} + 4442900 ) / 22800 \) Copy content Toggle raw display

Display \(\nu^j\) in terms of \(\beta_i\)

\(\nu\)\(=\) \( \beta_1 \) Copy content Toggle raw display
\(\nu^{2}\)\(=\) \( \beta_{2} + 13 \) Copy content Toggle raw display
\(\nu^{3}\)\(=\) \( \beta_{10} + \beta_{9} + 21\beta_1 \) Copy content Toggle raw display
\(\nu^{4}\)\(=\) \( \beta_{11} - 2\beta_{8} + \beta_{7} - 3\beta_{3} + 28\beta_{2} + 270 \) Copy content Toggle raw display
\(\nu^{5}\)\(=\) \( 31\beta_{10} + 34\beta_{9} - 3\beta_{6} - 7\beta_{5} + 5\beta_{4} + 498\beta_1 \) Copy content Toggle raw display
\(\nu^{6}\)\(=\) \( 45\beta_{11} - 75\beta_{8} + 65\beta_{7} - 122\beta_{3} + 745\beta_{2} + 6364 \) Copy content Toggle raw display
\(\nu^{7}\)\(=\) \( 850\beta_{10} + 950\beta_{9} - 70\beta_{6} - 320\beta_{5} + 252\beta_{4} + 12401\beta_1 \) Copy content Toggle raw display
\(\nu^{8}\)\(=\) \( 1520\beta_{11} - 2190\beta_{8} + 2760\beta_{7} - 4170\beta_{3} + 19755\beta_{2} + 158105 \) Copy content Toggle raw display
\(\nu^{9}\)\(=\) \( 22615\beta_{10} + 25085\beta_{9} - 770\beta_{6} - 11030\beta_{5} + 9690\beta_{4} + 316415\beta_1 \) Copy content Toggle raw display
\(\nu^{10}\)\(=\) \( 46375\beta_{11} - 59500\beta_{8} + 99375\beta_{7} - 135055\beta_{3} + 524500\beta_{2} + 4030160 \) Copy content Toggle raw display
\(\nu^{11}\)\(=\) \( 597125\beta_{10} + 650000\beta_{9} + 13625\beta_{6} - 344375\beta_{5} + 333805\beta_{4} + 8182090\beta_1 \) Copy content Toggle raw display

Display \(\beta_i\) in terms of \(\nu^j\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Copy content comment:embeddings in the coefficient field
 
Copy content gp:mfembed(f)
 
Label   \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
1.1
−5.22275
−4.87111
−3.53319
−2.97651
−1.24842
−1.04572
1.04572
1.24842
2.97651
3.53319
4.87111
5.22275
−5.22275 0 19.2771 0 0 26.2275 −58.8976 0 0
1.2 −4.87111 0 15.7277 0 0 10.1198 −37.6425 0 0
1.3 −3.53319 0 4.48344 0 0 −0.236432 12.4247 0 0
1.4 −2.97651 0 0.859630 0 0 −14.6792 21.2534 0 0
1.5 −1.24842 0 −6.44144 0 0 22.2415 18.0290 0 0
1.6 −1.04572 0 −6.90647 0 0 −23.4412 15.5880 0 0
1.7 1.04572 0 −6.90647 0 0 23.4412 −15.5880 0 0
1.8 1.24842 0 −6.44144 0 0 −22.2415 −18.0290 0 0
1.9 2.97651 0 0.859630 0 0 14.6792 −21.2534 0 0
1.10 3.53319 0 4.48344 0 0 0.236432 −12.4247 0 0
1.11 4.87111 0 15.7277 0 0 −10.1198 37.6425 0 0
1.12 5.22275 0 19.2771 0 0 −26.2275 58.8976 0 0
\(n\): e.g. 2-40 or 80-90
Embeddings: e.g. 1-3 or 1.12
Significant digits:
Format:

Atkin-Lehner signs

\( p \) Sign
\(3\) \( +1 \)
\(5\) \( -1 \)

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
5.b even 2 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1125.4.a.q ✓ 12
3.b odd 2 1 1125.4.a.r yes 12
5.b even 2 1 inner 1125.4.a.q ✓ 12
15.d odd 2 1 1125.4.a.r yes 12
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
1125.4.a.q ✓ 12 1.a even 1 1 trivial
1125.4.a.q ✓ 12 5.b even 2 1 inner
1125.4.a.r yes 12 3.b odd 2 1
1125.4.a.r yes 12 15.d odd 2 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{4}^{\mathrm{new}}(\Gamma_0(1125))\):

\( T_{2}^{12} - 75T_{2}^{10} + 2040T_{2}^{8} - 24475T_{2}^{6} + 126325T_{2}^{4} - 223000T_{2}^{2} + 122000 \) Copy content Toggle raw display
\( T_{11}^{6} + 85T_{11}^{5} - 1025T_{11}^{4} - 175500T_{11}^{3} - 1206875T_{11}^{2} + 89959375T_{11} + 1257671875 \) Copy content Toggle raw display

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( T^{12} - 75 T^{10} + \cdots + 122000 \) Copy content Toggle raw display
$3$ \( T^{12} \) Copy content Toggle raw display
$5$ \( T^{12} \) Copy content Toggle raw display
$7$ \( T^{12} + \cdots + 230656250000 \) Copy content Toggle raw display
$11$ \( (T^{6} + 85 T^{5} + \cdots + 1257671875)^{2} \) Copy content Toggle raw display
$13$ \( T^{12} + \cdots + 23\!\cdots\!25 \) Copy content Toggle raw display
$17$ \( T^{12} + \cdots + 10\!\cdots\!25 \) Copy content Toggle raw display
$19$ \( (T^{6} - 76 T^{5} + \cdots - 34508207164)^{2} \) Copy content Toggle raw display
$23$ \( T^{12} + \cdots + 99\!\cdots\!25 \) Copy content Toggle raw display
$29$ \( (T^{6} + \cdots - 9926453109375)^{2} \) Copy content Toggle raw display
$31$ \( (T^{6} + \cdots + 5444753644229)^{2} \) Copy content Toggle raw display
$37$ \( T^{12} + \cdots + 13\!\cdots\!25 \) Copy content Toggle raw display
$41$ \( (T^{6} + \cdots + 81872034750000)^{2} \) Copy content Toggle raw display
$43$ \( T^{12} + \cdots + 18\!\cdots\!25 \) Copy content Toggle raw display
$47$ \( T^{12} + \cdots + 16\!\cdots\!25 \) Copy content Toggle raw display
$53$ \( T^{12} + \cdots + 21\!\cdots\!00 \) Copy content Toggle raw display
$59$ \( (T^{6} + \cdots + 11\!\cdots\!25)^{2} \) Copy content Toggle raw display
$61$ \( (T^{6} + \cdots - 350580737229556)^{2} \) Copy content Toggle raw display
$67$ \( T^{12} + \cdots + 38\!\cdots\!00 \) Copy content Toggle raw display
$71$ \( (T^{6} + \cdots + 35\!\cdots\!00)^{2} \) Copy content Toggle raw display
$73$ \( T^{12} + \cdots + 32\!\cdots\!00 \) Copy content Toggle raw display
$79$ \( (T^{6} + \cdots - 24\!\cdots\!04)^{2} \) Copy content Toggle raw display
$83$ \( T^{12} + \cdots + 14\!\cdots\!00 \) Copy content Toggle raw display
$89$ \( (T^{6} + \cdots + 11\!\cdots\!00)^{2} \) Copy content Toggle raw display
$97$ \( T^{12} + \cdots + 18\!\cdots\!00 \) Copy content Toggle raw display
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